Observation of boundary induced chiral anomaly bulk states and their transport properties

The most useful property of topological materials is perhaps the robust transport of topological edge modes, whose existence depends on bulk topological invariants. This means that we need to make volumetric changes to many atoms in the bulk to control the transport properties of the edges in a sample. We suggest here that we can do the reverse in some cases: the properties of the edge can be used to induce chiral transport phenomena in some bulk modes. Specifically, we show that a topologically trivial 2D hexagonal phononic crystal slab (waveguide) bounded by hard-wall boundaries guarantees the existence of bulk modes with chiral anomaly inside a pseudogap due to finite size effect. We experimentally observed robust valley-selected transport, complete valley state conversion, and valley focusing of the chiral anomaly bulk states (CABSs) in such phononic crystal waveguides. The same concept also applies to electromagnetics.


The band dispersion of the PC slab with different layer numbers (integer)
shows the evolution of the CABS for PC strips with different layer numbers, from a minimum of 2 layers to a wide strip. If the width of PC remains an even (or odd) number layer, the dispersion stays the same, while the group velocities switch their signs when the layer thickness number changes from even to odd. We note that the frequency range of the CABSs is reduced with increasing width of the PC waveguide. In the limit of a very wide waveguide, the pseudo-gap with gradually shrink and the waveguide dispersion is equivalent to the projected band structure of the bulk phononic crystal, but the chiral anomaly remains at the Dirac point as long as the slab is of finite thickness and bounded by hard-wall boundary conditions.

The theoretical explanation and simulation results on the influence of the boundary conditions.
Here we use a boundary matrix method [see for example, Ref. 28,29] to analytically solve the wavefunctions and dispersions of all the states near the K point (the case in K' point is similar, since the system has time-reversal symmetry).
The physics near the K valley is effectively characterized by the 2D Dirac Hamiltonian: where denotes the parallel wavevector relative to the K point. The bulk eigenstates at the K valleysatisfy the static 2D Dirac equation Since the system has rigid boundaries at = 0 and = , the wave function bounded by the rigid boundaries can be expressed as the superposition of the two linearly independent bulk eigenstates at each : where the two bulk eigenstates | 1 ( , )⟩ = ( ( − ) ) exp( ) and | 2 ( , )⟩ = ( ( + ) ) exp(− ( − )) with = √ 2 − 2 / 2 are solved from Eq. (S2).
Hence, we can get Next we will examine the effect of bulk state when changing boundaries. The effect of the boundary tuning at = 0 and = can be represented as By virtue of these constraints, the two the boundary matrixes M 1 and M 2 can be expressed as M 1 ( 1 ) = sin 1 + cos 1 , M 2 ( 2 ) = sin 2 + cos 2 , where 1 and 2 are determined by the lower (upper) boundary condition of the real system.
This equation has nonzero solutions when the determinant of the matrix R vanishes: det ( , , 1 , 2 ) = 0. (S12) For a fixed boundary condition, 1 and 2 are determined, and only and is unknown in Eqs. (S11) (This equation is hard to get analytical solution, but their relationship is deterministic). Hence, the relationship between and can be obtained: Especially, when 1 = 2 = π 2 , we have ̂1 =̂2 = is coincident with the mirror-y operator of the Dirac Hamiltonian (see main text) and | ( , 0)⟩ = | ( , )⟩ ∝ ( 1 1 ) indicates the boundary conditions select the bulk even modes with = , which corresponds to the band structure in Fig. 1c. The existence of boundaries makes additional boundary potential to be applied in the vicinity of the boundaries. We take the additional boundary potential 1 ( ) applied in the vicinity of the bottom edge ( 1 (0 < < 0 ) ≠ 0 and 1 ( ≥ 0 ) = 0 with 0 → 0) as an example.
The modified Hamiltonian is The transfer matrix T( 1 , 2 ) are introduced to connect the wavefunctions | ( )⟩ from 1 to 2 .
If no boundary potential is applied, the transfer matrix is 0 . When ≥ 0 , the transfer matrix can be expressed as and the bulk states can be written as According to Eqs. (S5, S18), the following relationship is established is the equivalent boundary matrix and satisfy Comparing Eqs. S19 and S20, we can get Since 0 is very small compared with wavelength, the Eq. (S16) can be simplified to Combining Eqs. (S21) and (S22), one obtains a simple form of the effective boundary matrix which is identical to the original boundary matrix up to a parameter shift 1 : The additional boundary potential 2 ( ) applied in the vicinity of the upper edge ( → ) has the similar effect: These verify that modifying the boundary matrix can simulate the effect of adding additional boundaries. Hence when two additional boundary potentials 1 ( ) and 2 ( ) are added, the energy will the following change: = ( , 1 + 1 , 2 + 2 ).
Moreover, we show the influence of the boundary truncation positions on the band structure (simulated by COMSOL) in Fig. S2. In the process of changing the boundary truncation positions at two outmost layers, we ensure that the PC waveguide maintains mirror-y symmetry. Fig. S2a shows the band structure of the PC slab with a width varying from 10 to 12 layers (normalized by This experimentally verifies the robust properties of CABSs over a wide frequency range inside the pseudo-gap.

Details of the modified waveguide samples
The structural details of the modified waveguides in Fig. 4b,c are shown in Fig. S5a,b respectively.

The influence about the different cutting boundaries
The 60-bend waveguide, as shown in Fig. 4a, can be seen as consisting of two waveguides in domain 1 and 3, and the junction of the two waveguides is the coupling region. The sound wave cannot pass through the 60-bend waveguide smoothly. In order to change this state, we need to make some modifications in the coupling region.
Since two waveguides (domain 1 and 3) are symmetric about the middle line of the coupler (i.e. the line connecting the two corners). To maintain the symmetry, we truncate the coupling region along an angle of 30 degrees from the x-direction. At the same time, we know that the transmittance of the waveguide is related to its width.
Therefore, under the premise of ensuring symmetry, we cut different widths in coupling region, as shown in Fig. S6a, to see which one has the highest transmission.
And we find the truncated boundary 2 has the best results, the transmission of the 60-bend waveguide (green color in Fig. S6b) closes to 1 in the shaded frequency range. As a result, the sound wave can pass through the 60 degree-bend in Fig. 4c, and the reflection is highly suppressed (which is proved by the Fourier transformation in domain 1 where only K valley states are excited), after the boundary modification.

The CABSs in electromagnetic wave systems
The concept of creating CABSs by modifying the boundary in acoustic systems can also be applied to other systems. Here, we give two examples based on electromagnetic wave systems in microwave frequency and light wave frequency, respectively.
In microwave frequency, we consider a hexagonal dielectric photonic crystal For a waveguide of finite width, a pseudo-gap will appear at the K/K' point, with the size of the gap depending on the width, as in the case of acoustics. A chiral anomaly (a one-way mode in one valley) will appear in the pseudo-gap due to the boundary condition, as shown in the text.
For light wave frequencies, we consider a dielectric slab with a hexagonal array of air holes (150 nm lattice constant and 120 nm diameters) in air, as shown in Fig. S8a.
Open boundary conditions were applied at the top and bottom boundary of the waveguide, while the period boundary conditions were assumed at left and right, as shown in Fig. S8b. The material parameters of the dielectric slab are represented by the permittivity = 12 and permeability = 1. The wave guiding phenomenon in the light wave frequency is similar to the phenomenon in the acoustic system and microwave frequencies. This is because the interface between the slab with a high dielectric constant ( = 12 ) and the air has similar effect to the hard boundary in acoustic system.
These phenomena show that it is also feasible to generate CABSs by modifying the boundary in electromagnetic wave systems.